On the Accuracy of the Normal Approximation to Compound Poisson Distributions

Abstract

The value of the asymptotically exact constant is found in the analogue of the Berry--Esseen inequality for Poisson random sums of independent identically distributed random variables $X_1,X_2,\ldots$ possessing the third order moments. Moreover, for the uniform distance $\Delta_\lambda$ between the distribution function of the standard normal law and that of the centered and normalized random sum $S_\lambda=X_1+\cdots+X_{N_\lambda}$, ${N_\lambda}$ following the Poisson distribution with parameter $\lambda>0$ and being independent of $X_1,X_2,\ldots,$ the estimate $ \Delta_\lambda\le \frac{2\ell_\lambda}{3\sqrt{2\pi}} + 0.5\cdot\ell_\lambda^2< 0.2660\cdot\ell_\lambda+ 0.5\cdot\ell_\lambda^2$, where $\ell_\lambda = \frac{\E|X_1|^3}{\sqrt\lambda(\E X_1^2)^{3/2}}, $ is proved. It is demonstrated that this estimate is unimprovable regarding the factor $2/(3\sqrt{2\pi})=0.2659\ldots$ at $\ell_\lambda$. For the case of symmetric distribution of $X_1$ an improved bound $ \Delta_\lambda\le \frac{1+2\varkappa}{2\sqrt{2\pi}}\,\ell_\lambda + 0.4\cdot \ell_\lambda^2< 0.2391\cdot\ell_\lambda + 0.4\cdot \ell_\lambda^2 $ is obtained, where $\varkappa=\sup_{x>0}(\cos x-1+x^2/2)/x^3=0.0991\ldots\,.$ It is also shown that the value of the factor at $\ell_\lambda$ in this estimate cannot be made less than $(2\sqrt{2\pi})^{-1}=0.1994\ldots\,.$ Similar estimates are obtained under weakened moment conditions ${\bf E}|X_1|^{2+\delta}<\infty$ for some $0<\delta<1.$